which is invertible for generic \(\phi \) . It is straightforward to check that the so defined \(4\times 4\)  matrix-valued function \(\mathcal {X}\)  solves the \(\mathcal {X}\) -Riemann–Hilbert problem, RH-\(\mathcal {X}\) 1–RH-\(\mathcal {X}\) 4 with \(w \equiv 0\) , and \(\phi (z)\)  replaced by \(z^{1-r}\phi (z)\) . Take now \(r=0\) . That is, let us consider the standard orthogonal polynomials
on the circle with the weight \(\phi (z)\)  having zero winding number. Then, from the standard \(2\times 2\) 
Riemann–Hilbert formalism [1]}, we know everything about the asymptotic behavior of the corresponding orthogonal polynomials \(\mathcal {P}_n(z)\)  and hence we know asymptotic solution of the \(\mathcal {X}\) -Riemann–Hilbert problem corresponding
to \(D_n(\phi ,0;0,0)\) . And, we know this in spite of the fact that the approach developed in the body of this work can not be satisfactorily used for the case \(r=s=0\) . We believe that this observation might entail a useful hint on how to modify our Riemann–Hilbert approach to deal with general values of \(r\)  and \(s\) .
